If you wanted the answer to part 3, i have typed it out below:

1. expansion of (x+1)^n.(x+1)^n, and letting x=1:

(nC0 + nC1 + nc2 + ... + nCn)(nC0 + nC1 + nc2 + ... + nCn)

2. sum of coefficients of the x^(n+1) terms is:

(nC1)(nCn) + (nC2)(nCn-1) + (nC3)(nCn-2) + ... + (nCn)(nCn-1)

= (nC1)(nC0) + (nC2)(nC1) + (nC3)(nC2) + ... + (nCn)(nCn-1)

3. expansion of (1 + x)^2n is:

2nC0 + 2nC1(x) + 2nC2(x^2) + ... + 2nCn(x^n) + __2nC(n+1).(x^(n+1))__ +...

4. coefficient of the x^(n+1) term:

(2n)C(n+1) = (2n)!/(2n - n - 1)!(n + 1)!

= (2n)!/(n - 1)! (n + 1)!

:. (nC0)(nC1) + (nC1)(nC2) + (nC2)(nC3) + .... + (nC(n-1))(nCn) = (2n)! / (n-1)!(n+1)! as required